In mathematics, an inner product space or a Hausdorff preHilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors, often denoted using angle brackets (as in
⟨
a
,
b
⟩
{\displaystyle \langle a,b\rangle }
). Inner products allow the rigorous introduction of intuitive geometrical notions, such as the length of a vector or the angle between two vectors. They also provide the means of defining orthogonality between vectors (zero inner product). Inner product spaces generalize Euclidean spaces (in which the inner product is the dot product, also known as the scalar product) to vector spaces of any (possibly infinite) dimension, and are studied in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898.An inner product naturally induces an associated norm, (

x

{\displaystyle x}
and

y

{\displaystyle y}
are the norms of
x
{\displaystyle x}
and
y
{\displaystyle y}
in the picture), which canonically makes every inner product space into a normed vector space. If this normed space is also a Banach space then the inner product space is called a Hilbert space. If an inner product space
(
H
,
⟨
⋅
,
⋅
⟩
)
{\displaystyle (H,\langle \,\cdot \,,\,\cdot \,\rangle )}
is not a Hilbert space then it can be "extended" to a Hilbert space
(
H
¯
,
⟨
⋅
,
⋅
⟩
H
¯
)
,
{\displaystyle \left({\overline {H}},\langle \,\cdot \,,\,\cdot \,\rangle _{\overline {H}}\right),}
called a completion. Explicitly, this means that
H
{\displaystyle H}
is linearly and isometrically embedded onto a dense vector subspace of
H
¯
{\displaystyle {\overline {H}}}
and that the inner product
⟨
⋅
,
⋅
⟩
H
¯
{\displaystyle \langle \,\cdot \,,\,\cdot \,\rangle _{\overline {H}}}
on
H
¯
{\displaystyle {\overline {H}}}
is the unique continuous extension of the original inner product
Hi,
from Penrose book "The Road to Reality" it seems to me inner product and dot/scalar product are actually different things.
Given a vector space ##V## an inner product ## \langle .  . \rangle## is defined between elements (i.e. vectors) of the vector space ##V## itself. Differently...
In simple Euclidean space: From trig, we have , for u and v separated by angle Θ, the length of the projection of u onto v is ucosΘ; then from one definition of the dot product Θ=arcos(u⋅v/(u⋅v)); putting them together, I get the length of the projection of u onto v is u⋅v/v.
Then I...
Given an orthonormal basis ##\{e_1,\ldots, e_n\}## in a complex inner product space ##V## of dimension ##n##, show that if ##v_1,\ldots, v_n\in V## such that ##\sum_{j = 1}^n \v_j\^2 < 1##, then ##\{v_1 + e_1,\ldots, v_n + e_n\}## is a basis for ##V##.
Hi
In QM the inner product satisfies < a  a > ≥ 0 with equality if and only if a = 0.
Is this positive definite or positive semidefinite because i have seen it described as both
Thanks
Inner product is a generalization of the dot product on spaces other than Euclidean and for vectors it is defined in the same way as the dot product. If we have two vectors $v$ and $w$, than their inner product is: $$\langle vw\rangle = v_1w_1 + v_2w_2 + ...+v_nw_n $$
where $v_1,w_1...
##\langle T(f), g \rangle = \int_{0}^{1} \int_{0}^{x} f(t) dt ~ g(t) dt##
As ##\int_{0}^{x} f(t) dt## will be a function in ##x##, therefore a constant w.r.t. ##dt##, we have
##\langle T(f), g \rangle = \int_{0}^{x} f(t) dt ~ \int_{0}^{1} g(t) dt##
##\langle f, T(g)\rangle = \int_{0}^{1} f(t)...
Hi,
I was thinking about the following problem, but I couldn't think of any conclusive reasons to support my idea.
Question:
Let us imagine that we have two vectors ## \vec{a} ## and ## \vec{b} ## and they point in similar directions, such that the innerproduct is evaluated to be a +ve...
I'm learning Linear Algebra by self and I began with Apsotol's Calculus Vol 2. Things were going fine but in exercise 1.13 there appeared too many questions requiring a strong knowledge of Real Analysis. Here is one of it (question no. 14)
Let ##V## be the set of all real functions ##f##...
Suppose we have V, a finitedimensional complex vector space with a Hermitian inner product. Let T: V to V be an arbitrary linear operator, and T^* be its adjoint.
I wish to prove that T is diagonalizable iff for every eigenvector v of T, there is an eigenvector u of T^* such that <u, v> is...
We denote a scalar product of two vectors ##a, b## in Hilbert space ##H## as $(a,b)$.
In Bra Ket notation, we denote a vector a in Hilbert space as ##a\rangle##. Also we say that bras belong to the dual space ##H##∗ .
So Bras are linear transformations that map kets to a number.
Then it...
In https://mathworld.wolfram.com/InnerProduct.html, it states
"Every inner product space is a metric space. The metric is given by
g(v,w)= <vw,vw>."
In https://en.wikipedia.org/wiki/Inner_product_space , on the other hand,
"As for every normed vector space, an inner product space is a metric...
The unormalised plane wave solution is given as ##u_{\vec{k}}=e^{i\vec{k}\cdot\vec{x}i\omega t}##. I want to show that ##(u_{\vec{k}},u^{*}_{\vec{k}'})=0##. However, I don't seem to be able to get the answer through direct calculation. Any hints on how to obtain the answer?
This is a very elementary question, from the beginnings of quantum mechanics.
For simplicity, I refer to a finite case with pure states.
If I understand correctly, the spectrum of an observable is the collection of eigenvalues formed by the inner product of states and hence equal to...
b)
c and d):
In c) I say that ##L_h## is only self adjoint if the imaginary part of h is 0, is this correct?
e) Here I could only come up with eigenvalues when h is some constant say C, then C is an eigenvalue. But I' can't find two.Otherwise does bd above look correct?
Thanks in advance!
hi guys
i was thinking about the inner product we choose in quantum mechanics to map the elements inside the hilbert space to real number which is given by :
$$\int^{∞}_{∞}\psi^{*}\psi\;dV$$
or in some cases we might introduce a weight function dependent on the wave functions i have , it seems...
A coordinate system with the coordinates s and t in R^2 is defined by the coordinate transformations: s = y/y_0 and t=y/y_0  tan(x/x_0) , where x_0 and y_0 are constants.
a) Determine the area that includes the point (x, y) = (0, 0) where the coordinate system
is well defined. Express the...
Summary:: Inner Product Spaces, Orthogonality.
Hi there,
This my first thread on this forum :)
I encountered the above problem in Schaum’s Outlines of Linear Algebra 6th Ed (2017, McGrawHill) Chapter 7  Inner Product Spaces, Orthogonality.
Using some particular values for u and v, I...
In the section 82 dealing with resolving the state vectors, we learn that
\phi \rangle =\sum_i C_i  i \rangle
and the dual vector is defined as
\langle \chi  =\sum_j D^*_j \langle j Then, the an inner product is defined as
\langle \chi  \phi \rangle =\sum_{ij} D^*_j C_i \langle j  i...
Hi everyone,
I was attempting the following past paper question below:
I have found a value for the coefficient c and I think I have calculated the inner product of <xx>. I've attached my workings below. But I'm not sure what to do next to answer the last part of the question which asks...
(scroll to bottom for problem statement)
Hello,
I am wondering if someone could break down the problem statement in simpler terms (not so mathy).
I am struggling with understanding what is being asked.
I will try to break it down to the best of my ability
Problem statement:Consider the inner...
$$<p_1 p_2p_A p_B> = \sqrt{2E_1 2E_2 2E_A 2E_B}<0a_1 a_2 a_{A}^{\dagger} a_{B}^{\dagger} 0>$$ $$=2E_A2E_B(2\pi)^6(\delta^{(3)}(p_Ap_1)\delta{(3)}(p_Bp_2) + \delta^{(3)}(p_Ap_2)\delta^{(3)}(p_Bp_1))$$
The identity above seemed easy, until I tried to prove it. I figured I could work this...
Let <x, x>=3x_{1}^2+2x_{2}^2+x_{3}^24x_{1}x_{2}2x_{1}x_{3}+2x_{2}x_{3} be a quadratic form in V=R, where x=x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3} (in the base {e_{1},e_{2},e_{3}}.
Find the inner product corresponding to this quadratic form.
Is this that easy that you have to change '' second''...
I need help to know if I'm on the right track:
Prove/Disprove the following:
Let u ∈ V . If (u, v) = 0 for every v ∈ V such that v ≠ u, then u = 0.
(V is a vectorspace)
I think I need to disprove by using v = 0, however I'm not sure.
I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ...
I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ...
I need some help to confirm my thinking on Proposition...
I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ...
I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ...
I need some help to fully understand the proof of...
Problem:
Prove that for any $x \in R^n$ and any $0<p<\infty$
$\int_{S^{n1}} \rvert \xi \cdot x \rvert^p d\sigma(\xi) = \rvert x \rvert^p \int_{S^{n1}} \rvert \xi_1 \rvert^p d\sigma(\xi)$,
where $\xi \cdot x = \xi_1 x_1 + ... + \xi_n x_n$ is the inner product in $R^n$.
Some thinking...
I...
So say our inner product is defined as ##\int_a^b f^*(x)g(x) dx##, which is pretty standard. For some operator ##\hat A##, do we then have ## \langle \hat A ψ  \hat A ψ \rangle = \langle ψ  \hat A ^* \hat A  ψ \rangle = \int_a^b ψ^*(x) \hat A ^* \hat A ψ(x) dx##? This seems counterintuitive...
Hi everyone.
Yesterday I had an exam, and I spent half the exam trying to solve this question.
Show that ##\left\langle\Psi\left(\vec{r}\right)\right\hat{p_{y}^{2}}\left\phi\left(\vec{r}\right)\right\rangle =\left\langle...
Hey, I am currently reading over the linear algebra section of the "introduction to quantum mechanics" by Griffiths, in the Inner product he notes: "The inner product of two vector can be written very neatly in terms of their components: <aB>=a1* B1 + a2* B ... " He also took upon the...
In this video, at around 37:10 he is explaining the orthogonality of spherical harmonics. I don't understand his explanation of the \sin \theta in the integrand when taking the inner product. As I interpret this integral, we are integrating these two spherical harmonics over the surface of a...
Homework Statement
Let ##C \subset \mathbb{R}^n## a convex set. If ##x \in \mathbb{R}^n## and ##\overline{x} \in C## are points that satisfy ##x\overline{x}=d(x,C)##, proves that ##\langle x\overline{x},y\overline{x} \rangle \leq 0## for all ##y \in C##.
Homework Equations
By definition...
Homework Statement
Given ##a\neq b## vectors of ##\mathbb{R}^n##. Determine ##c## which lies in the line segment ##[a,b]=\{a+t(ba) ; t \in [0,1]\}##, such that ##c \perp (ba)##. Conclude that for all ##x \in [a,b]##, with ##x\neq c## it is true that ##c<x##.
Homework Equations
The first...
Homework Statement
Let ##x,y \in \mathbb{R^n}## not null vectors. If for all ##z \in \mathbb{R^n}## that is orthogonal to ##x## we have that ##z## is also orthogonal to ##y##, prove that ##x## and ##y## are multiple of each other.
Homework Equations
We can use that fact that ##<x ...
Wolfram says that an example of an inner product space is the vector space of real functions whose domain is an closed interval [a,b] with inner product ##\langle f, g\rangle = \int_a^b f(x) g(x) dx##. But ##1/x## is a real function, and ##\langle 1/x, 1/x\rangle## does not converge... So how is...
I am practicing old exams. I tried my best but looking at an old and a bit unreliable answer list, and i am not getting the same result.
Homework Statement
At time ##t=0## the nomralized harmonic oscialtor wavefunction is given by:
## \Psi(x,0) = \frac{1}{\sqrt{3}}(\psi_0(x) + \psi_1(x) + i...
I read from this page https://properphysics.wordpress.com/2014/06/09/anononsenseintroductiontospecialrelativitypart6/
that the basis vectors are the canonical basis vectors in any coordinate system. This seems to be wrong, because if that was the case the metric would be the identity...
Consider that a vector can be represented in two different basis. My question is do we need to normalize both basis before taking the inner product?
What motivates this question is because I found out that the inner product of a vector having components ##a,b## in the normalized polar basis of...
Homework Statement
Homework Equations
I am not sure. I have not seen the triangle inequality for inner products, nor the CauchySchwarz Inequality for the inner product. The only thing that my lecture notes and textbook show is the axioms for general inner products, the definition of norm...
Homework Statement
Prove that ##\vec {a} \cdot (\vec {b} \wedge \vec {C_r}) = \vec {a} \cdot \vec {b} \vec {C_r}  \vec {b} \wedge (\vec {a} \cdot \vec {C_r})##.
Note that ##\vec {a}## is a vector, ##\vec {b}## is a vector, and ##\vec {C_r}## is an rblade with ##r > 0##.
Also, the dot...
I am trying to follow modern QFT by Tom Banks and I am having an issue with literally the first equation.
He claims that beginning from ## p_1 , p_2, ... , p_k> \: = \: a^\dagger (p_1) a^\dagger (p_2) \cdots a^\dagger (p_k)0> ## with the commutation relation ##[a (p),a^\dagger (q)]_\pm \: =...
Hi, what is the physical meaning, or also the geometrical meaning of the inner product of two eigenvectors of a matrix?
I learned from the previous topics that a vectors space is NOT Hilbert space, however an inner product forms a Hilbert space if it is complete.
Can two eigenvectors which...
Hi, I am trying to prove that the eigevalues, elements, eigenfunctions or/and eigenvectors of a matrix A form a Hilbert space. Can one apply the inner product formula :
\begin{equation}
\int x(t)\overline y(t) dt
\end{equation}
on the x and y coordinates of the eigenvectors [x_1,y_1] and...
I'm reading a textbook on electromagnetism. It says that for two vector fields ##\textbf{F}(\textbf{r})##
and ##\textbf{G}(\textbf{r})## their inner product is defined as
##(\textbf{F},\textbf{G}) = \int \textbf{F}^{*}\cdot \textbf{G} \thinspace d^3\textbf{r}##
And that if ##\textbf{F}## is...
I am new to quantum mechanics and I have recently been reading Shankar's book. It was all good until I reached the idea of representing functions of continouis variable as kets for example f(x)>. The book just scraped off the definition of inner product in the discrete space case and refined it...
Hi,
I'm stuck on a problem from my quantum homework. I have to show <p1p2> = ∫(from 1 to 1) dx (p1*)(p2)
is a scalar product (p1 and p2 are single variable complex polynomials). I've figured out how to show that they satisfy linearity and positive definiteness, but I'm completely stuck on...
I started learning quantum, and I got a bit confused about inner and dot products.
I've attached 2 screenshots; 1 from Wikipedia and 1 from an MIT pdf I found online.
Wikipedia says that a.Dot(b) when they're complex would be the sum of aibi where b is the complex conjugate.
The PDF from MIT...
Homework Statement
Homework Equations
The Attempt at a Solution
[/B]
I could show that the first of the three properties are valid for any value of a,b,c but I couldn’t find a way to show the forth one.
Follow all the procedures I already did:
Homework Statement
Let ##V## be a vector space equipped with an inner product ##\langle \cdot, \cdot \rangle##. If ##\langle x,y \rangle = \langle x, z\rangle## for all ##x \in V##, then ##y=z##.
Homework EquationsThe Attempt at a Solution
Here is my attempt. ##\langle x,y \rangle = \langle x...